CSE 473: Artificial Intelligence - PowerPoint Presentation

Slide1

CSE 473: Artificial Intelligence

Constraint Satisfaction

Daniel Weld

Slides

adapted from Dan Klein, Stuart

Russell, Andrew Moore & Luke

Zettlemoyer

Slide2

Recap:

Search Problem

States

configurations

of the

world

Successor function:

function

from states

to lists

of

triples

(state

, action,

cost)

Start state

Goal

test

Slide3

General Tree Search Paradigm

3

function tree-search(root-node)

fringe 

successors

(root-node)

while (

notempty

(fringe) ) {node  remove-first(fringe) state  state(node) if goal-test(state) return solution(node) fringe  insert-all(successors(node),fringe) } return failureend tree-search

Fringe managed asStackQueuePriority Queue

Depth first

Breadth first

Best first, uniform cost, greedy, A*

Slide4

4

Space of Search

Strategies

Blind

Search

DFS, BFS, IDS

Informed

Search

Uniform cost, greedy, A*, IDA*Constraint SatisfactionBacktracking=DFS, FC, k-consistencyAdversary SearchSystematic / Stochastic (“Local”) 

 

Fri

Fri

Slide5

5

Constraint Satisfaction

Kind of

search

in which

States are

factored

into sets of variables

Search = assigning values to these variablesGoal test is encoded with constraints  Gives structure to search space Exploration of one part informs othersBacktracking-style algorithms workBut other techniques add speedPropagationVariable orderingPreprocessing

Slide6

Example: N-Queens

Formulation

as search

States

Operators

Start State

Goal Test

Slide7

Constraint Satisfaction Problems

Standard search problems:

State is a “black box”: arbitrary data structure

Goal test: any function over states

Successor function can be anything

Simple example of a

formal representation language

Allows

more powerful

search algorithms

Constraint satisfaction problems (CSPs):

A special subset of search problems

State is defined by

variables

X

i

with values from a

domain

D

(often

D

depends on

i

)

Goal test is a

set of constraints

specifying allowable combinations of values for subsets of variables

Slide8

Example: N-Queens

CSP Formulation

1:

Variables:

Domains:

Constraints

X

ij

+

X

ik

≤ 1

X

ij

+

X

kj

≤ 1

X

ij

+

X

i+k,j+

k

≤ 1

X

ij

+

X

i+k,j-

k

≤ 1

Slide9

Example: N-Queens

CSP Formulation

1:

Variables:

Domains:

Constraints

Slide10

Example: N-Queens

Formulation 1.5:

Variables:

Domains:

Constraints:

… there’s an even better way! What is it?

Slide11

Example: N-Queens

Formulation 2:

Variables:

Domains:

Constraints:

Implicit:

Explicit:

-or-

Slide12

Example: Map-Coloring

Variables:

Domain:

Constraints: adjacent regions must have different colors

Solutions are assignments satisfying all constraints, e.g.:

Slide13

Constraint Graphs

Binary CSP: each constraint relates (at most) two variables

Binary constraint graph: nodes are variables, arcs show constraints

General-purpose CSP algorithms use the graph structure to speed up search. E.g., Tasmania is an independent subproblem!

Slide14

Example: Sudoku

Variables:

Domains:

Constraints:

9-way alldiff for each row

9-way alldiff for each column

9-way alldiff for each region

Each (open) square

{1,2,…,9}

Slide15

Example: Cryptarithmetic

Variables (circles):

Domains:

Constraints (boxes):

Slide16

Example: The Waltz Algorithm

The Waltz algorithm is for interpreting line drawings of solid polyhedra

An early example of a computation posed as a CSP

Look at all intersections

Adjacent intersections impose constraints on each other

?

Slide17

Waltz on Simple Scenes

Assume all objects:

Have no shadows or cracks

Three-faced vertices

“General position”: no junctions change with small movements of the eye.

Then each line on image is one of the following:

Boundary line (edge of an object) (

→

) with right hand of arrow denoting “solid” and left hand denoting “space”Interior convex edge (+)Interior concave edge (-)

Slide18

Legal Junctions

Only certain junctions are physically possible

How can we formulate a CSP to label an image?

Variables: vertices

Domains: junction labels

Constraints: both ends of a line should have the same label

x

y

(x,y) in

,

, …

Slide19

Varieties of CSPs

Discrete Variables

Finite domains

Size

d

means

O(

d

n) complete assignmentsE.g., Boolean CSPs, including Boolean satisfiability (NP-complete)Infinite domains (integers, strings, etc.)E.g., job scheduling, variables are start/end times for each jobLinear constraints solvable, nonlinear undecidableContinuous variablesE.g., start/end times for Hubble Telescope observationsLinear constraints solvable in polynomial time by LP methods

Slide20

Varieties of Constraints

Varieties of Constraints

Unary constraints involve a single variable (equiv. to shrinking domains):

Binary constraints involve pairs of variables:

Higher-order constraints involve 3 or more variables:

e.g., cryptarithmetic column constraints

Preferences (soft constraints):

E.g., red is better than greenOften representable by a cost for each variable assignmentGives constrained optimization problems(We’ll ignore these until we get to Bayes’ nets)

Slide21

Real-World CSPs

Assignment problems: e.g., who teaches what class

Timetabling problems: e.g., which class is offered when and where?

Hardware

configuration

Gate assignment in airports

Transportation scheduling

Factory scheduling

Fault diagnosis… lots more!Many real-world problems involve real-valued variables…

Slide22

Standard Search Formulation

Standard search formulation of CSPs (incremental)

Start

with

straightforward approach

, then

improve

States are defined by the values assigned so far

Initial state: the empty assignment, {}Successor function: assign a value to an unassigned variableGoal test: the current assignment is complete and satisfies all constraints

Slide23

Search Methods

What does BFS do?

What does DFS do?

Slide24

DFS, and BFS would be much worse!

Slide25

Backtracking Search

Idea 2: Only allow legal assignments at each point

I.e. consider only values which do not conflict previous assignments

Might have to do some computation to figure out whether a value is ok

“Incremental goal test”

Depth-first search for CSPs with these two improvements is called

backtracking search

(useless name, really)

Backtracking search is the basic uninformed algorithm for CSPsCan solve n-queens for n ≈ 25

Idea 1: Only consider a single variable at each pointVariable assignments are commutative, so fix orderingI.e., [WA = red then NT = green] same as [NT = green then WA = red]

Only need to consider assignments to a single variable at each step

How many leaves are there?

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Backtracking Search

What are the choice points?

Slide27

Backtracking Example

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Backtracking

Slide29

Are we done?

Slide30

Improving Backtracking

General-purpose ideas give huge gains in speed

Ordering:

Which variable should be assigned next?

In what order should its values be tried?

Filtering: Can we detect inevitable failure early?

Structure: Can we exploit the problem structure?

Slide31

Forward Checking

Idea: Keep track of remaining legal values for unassigned variables (using immediate constraints)

Idea: Terminate when any variable has no legal values

WA

SA

NT

Q

NSW

V

Slide32

Forward Checking

Slide33

Are We Done?

Slide34

Constraint Propagation

Forward checking propagates information from assigned to adjacent unassigned variables, but doesn't detect more distant failures:

WA

SA

NT

Q

NSW

V

NT and SA cannot both be blue!

Why didn’t we detect this yet?

Constraint propagation

repeatedly enforces constraints (locally)

Slide35

Arc Consistency

Simplest form of propagation makes each arc

consistent

X

→

Y is consistent iff for

every

value x there is

some allowed y

WA

SA

NT

Q

NSW

V

If X loses a value, neighbors of X need to be rechecked!

Arc consistency detects failure earlier than forward checking

What’s the downside of arc consistency?

Can be run as a preprocessor or after each assignment

Slide36

Arc Consistency

Runtime: O(n

2

d

3

), can be reduced to O(n

2

d

2)… but detecting all possible future problems is NP-hard – why?[demo: arc consistency animation]

Slide37

Constraint

Propagation

Slide38

Are We Done?

Slide39

Limitations of Arc Consistency

After running arc consistency:

Can have one solution left

Can have multiple solutions left

Can have no solutions left (and not know it)

What went wrong here?

Slide40

K-Consistency*

Increasing degrees of consistency

1-Consistency (Node Consistency): Each single node’s domain has a value which meets that node’s unary constraints

2-Consistency (Arc Consistency): For each pair of nodes, any consistent assignment to one can be extended to the other

K-Consistency: For each k nodes, any consistent assignment to k-1 can be extended to the k

th

node.

Higher k more expensive to compute

(You need to know the k=2 algorithm)

Slide41

Strong K-Consistency

Strong k-consistency: also k-1, k-2, … 1 consistent

Claim: strong n-consistency means we can solve without backtracking!

Why?

Choose any assignment to any variable

Choose a new variable

By 2-consistency, there is a choice consistent with the first

Choose a new variable

By 3-consistency, there is a choice consistent with the first 2…Lots of middle ground between arc consistency and n-consistency! (e.g. path consistency)

Slide42

Ordering: Minimum Remaining Values

Minimum remaining values (MRV):

Choose the variable with the fewest legal values

Why min rather than max?

Also called “most constrained variable”

“Fail-fast” ordering

Slide43

Ordering: Degree Heuristic

Tie-breaker among MRV variables

Degree heuristic:

Choose the variable participating in the most constraints on remaining variables

Why most rather than fewest constraints?

Slide44

Ordering: Least Constraining Value

Given a choice of variable:

Choose the

least constraining value

The one that rules out the fewest values in the remaining variables

Note that it may take some computation to determine this!

Why least rather than most?

Combining these heuristics makes 1000 queens feasible

Slide45

Propagation with Ordering

Slide46

Problem Structure

Tasmania and mainland are independent subproblems

Identifiable as connected components of constraint graph

Suppose each subproblem has c variables out of n total

Worst-case solution cost is O((n/c)(d

c

)), linear in n

E.g., n = 80, d = 2, c =20

280 = 4 billion years at 10 million nodes/sec(4)(220) = 0.4 seconds at 10 million nodes/sec

Slide47

Tree-Structured CSPs

Choose a variable as root, order

variables from root to leaves such

that every node's parent precedes

it in the ordering

For i = n : 2, apply RemoveInconsistent(Parent(X

i

),X

i)For i = 1 : n, assign Xi consistently with Parent(Xi)Runtime: O(n d2)

Slide48

Tree-Structured CSPs

Theorem: if the constraint graph has no loops, the CSP can be solved in O(n d

2

) time!

Compare to general CSPs, where worst-case time is O(d

n

)

This property also applies to logical and probabilistic reasoning: an important example of the relation between syntactic restrictions and the complexity of reasoning.

Slide49

Nearly Tree-Structured CSPs

Conditioning: instantiate a variable, prune its neighbors' domains

Cutset conditioning: instantiate (in all ways) a set of variables such that the remaining constraint graph is a tree

Cutset size c gives runtime O( (d

c

) (n-c) d

2

), very fast for small c

Slide50

Iterative Algorithms for CSPs

Greedy and local methods typically work with “complete” states, i.e., all variables assigned

To apply to CSPs:

Allow states with unsatisfied constraints

Operators

reassign

variable values

Variable selection: randomly select any conflicted variable

Value selection by min-conflicts heuristic:Choose value that violates the fewest constraintsI.e., hill climb with h(n) = total number of violated constraints

Slide51

Example: 4-Queens

States: 4 queens in 4 columns (4

4

= 256 states)

Operators: move queen in column

Goal test: no attacks

Evaluation: h(n) = number of attacks

Slide52

Performance of Min-Conflicts

Given random initial state, can solve n-queens in almost constant time for arbitrary n with high probability (e.g., n = 10,000,000)

The same appears to be true for any randomly-generated CSP

except

in a narrow range of the ratio

Slide53

Summary

CSPs are a special kind of search problem:

States defined by values of a fixed set of variables

Goal test defined by constraints on variable values

Backtracking = depth-first search with one legal variable assigned per node

Variable ordering and value selection heuristics help significantly

Forward checking prevents assignments that guarantee later failure

Constraint propagation (e.g., arc consistency) does additional work to constrain values and detect inconsistencies

The constraint graph representation allows analysis of problem structureTree-structured CSPs can be solved in linear timeIterative min-conflicts is usually effective in practice

Slide54

© Daniel S. Weld

54Chinese Food as Search?

States?

Operators?

Start state?

Goal states?

Partially specified meals

Add, remove, change dishes

Null meal

Meal meeting certain conditions (rating?)

Slide55

© Daniel S. Weld

55Factoring States

Rather than state = meal

Model state’s (independent) parts, e.g.

Suppose every meal for n people

Has n dishes plus soup

Soup =

Meal 1 =

Meal 2 = …Meal n = Or… physical state =X coordinate =Y coordinate =

Slide56

© Daniel S. Weld

56Chinese Constraint

Network

Soup

Total Cost

< $30

Chicken

Dish

Vegetable

Rice

Seafood

Pork Dish

Appetizer

Must be

Hot&Sour

No

Peanuts

No

Peanuts

Not

Chow Mein

Not Both

Spicy

Slide57

© Daniel S. Weld

57CSPs in the Real World

Scheduling space shuttle repair

Airport gate assignments

Transportation Planning

Supply-chain management

Computer configuration

Diagnosis

UI optimizationEtc...

Slide58

© Daniel S. Weld

58Classroom Scheduling

Variables?

Domains (possible values for variables)?

Constraints?

Slide59

© Daniel S. Weld

59CSP as a search problem?

What are states?

(nodes in graph)

What are the operators?

(arcs between nodes)

Initial state?

Goal test?

Q

Q

Q

Slide60

© Daniel S. Weld

60Crosswords